| L(s) = 1 | + (0.291 − 1.65i)2-s + (−0.775 − 0.282i)4-s + (−0.865 + 0.726i)5-s + (3.67 − 1.33i)7-s + (0.987 − 1.70i)8-s + (0.949 + 1.64i)10-s + (−1.43 − 1.20i)11-s + (−0.127 − 0.721i)13-s + (−1.14 − 6.47i)14-s + (−3.80 − 3.19i)16-s + (−0.944 − 1.63i)17-s + (−1.37 + 2.37i)19-s + (0.876 − 0.318i)20-s + (−2.40 + 2.02i)22-s + (5.47 + 1.99i)23-s + ⋯ |
| L(s) = 1 | + (0.206 − 1.17i)2-s + (−0.387 − 0.141i)4-s + (−0.386 + 0.324i)5-s + (1.38 − 0.505i)7-s + (0.349 − 0.604i)8-s + (0.300 + 0.519i)10-s + (−0.432 − 0.362i)11-s + (−0.0352 − 0.200i)13-s + (−0.304 − 1.72i)14-s + (−0.951 − 0.798i)16-s + (−0.229 − 0.396i)17-s + (−0.314 + 0.544i)19-s + (0.195 − 0.0713i)20-s + (−0.513 + 0.430i)22-s + (1.14 + 0.415i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.01315 - 1.12959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01315 - 1.12959i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.291 + 1.65i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.865 - 0.726i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-3.67 + 1.33i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (1.43 + 1.20i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.127 + 0.721i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.944 + 1.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.37 - 2.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.47 - 1.99i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.923 - 5.23i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.25 + 0.458i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (1.69 + 2.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.311 - 1.76i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.85 - 3.23i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (1.60 - 0.584i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + (8.62 - 7.23i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (4.91 - 1.78i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.328 + 1.86i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.09 - 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.14 + 12.1i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.02 - 11.5i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.86 + 4.96i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.263 + 0.220i)T + (16.8 + 95.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.55823371686823706585364989333, −11.02504381817964198213300633038, −10.47899467490181066494766677067, −9.125324109057664717855016577570, −7.83909563329848244505917281804, −7.10030702720620041884363059407, −5.26589295007597544782340666260, −4.15198496051803634875847941350, −2.97167219110954363888102359523, −1.43961061496835071701722404354,
2.13863669683817358063535052656, 4.50279984907989737616451423903, 5.11069200464464995650337393093, 6.32281716933899131009432310023, 7.48415253212849858365633429258, 8.210715711670320638202790370183, 8.966198133658045417155003936400, 10.65561435366293719733377407471, 11.40727986878587954049784746630, 12.38631179122854305735853509286
